Gauss-Siedel Method

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Gauss-Siedel Method

• Industrial Engineering Majors
• Authors Autar Kaw
• http//numericalmethods.eng.usf.edu
• Transforming Numerical Methods Education for STEM
  Undergraduates

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Gauss-Seidel Method http//numericalmethod
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Gauss-Seidel Method
An iterative method.

• Basic Procedure
• Algebraically solve each linear equation for xi
• Assume an initial guess solution array
• Solve for each xi and repeat
• Use absolute relative approximate error after
  each iteration to check if error is within a
  pre-specified tolerance.

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Gauss-Seidel Method
Why?
The Gauss-Seidel Method allows the user to
control round-off error. Elimination methods
such as Gaussian Elimination and LU Decomposition
are prone to prone to round-off error. Also If
the physics of the problem are understood, a
close initial guess can be made, decreasing the
number of iterations needed.
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Gauss-Seidel Method
Algorithm
A set of n equations and n unknowns
If the diagonal elements are non-zero Rewrite
each equation solving for the corresponding
unknown ex First equation, solve for x1 Second
equation, solve for x2
. . .
. . .
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Gauss-Seidel Method
Algorithm
Rewriting each equation
From Equation 1 From equation 2 From
equation n-1 From equation n
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Gauss-Seidel Method
Algorithm
General Form of each equation
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Gauss-Seidel Method
Algorithm
General Form for any row i
How or where can this equation be used?
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Gauss-Seidel Method
Solve for the unknowns
Use rewritten equations to solve for each value
of xi. Important Remember to use the most recent
value of xi. Which means to apply values
calculated to the calculations remaining in the
current iteration.
Assume an initial guess for X
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Gauss-Seidel Method
Calculate the Absolute Relative Approximate Error
So when has the answer been found? The
iterations are stopped when the absolute relative
approximate error is less than a prespecified
tolerance for all unknowns.
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Example Production Optimization
To find the number of toys a company should
manufacture per day to optimally use their
injection-molding machine and the assembly line,
one needs to solve the following set of
equations. The unknowns are the number of toys
for boys, x1, number of toys for girls, x2, and
the number of unisexual toys, x3.
Find the values of x1, x2,and x3 using the
Gauss-Seidel Method
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Example Production Optimization
The system of equations is


Initial Guess Assume an initial guess of
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Example Production Optimization
Rewriting each equation


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Example Production Optimization
The Equation for x3 is divided by 0 which is
undefined. Therefore the order of the equations
will need to be reordered. Equation 3 and
equation 1 will be switched. By switching
equations 3 and 1, the matrix will also become
diagonally dominant. The system of equations
becomes


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Example Production Optimization
Rewriting each equation


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Example Production Optimization
Applying the initial guess and solving for xi
Initial Guess


When solving for x2, how many of the initial
guess values were used?
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Example Production Optimization
Finding the absolute relative approximate error

At the end of the first iteration

The maximum absolute relative approximate error
is 61.138
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Example Production Optimization
Iteration 2
Using from Iteration
1, the values of xi are found




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Example Production Optimization
Finding the absolute relative approximate error
At the end of the second iteration
The maximum absolute relative approximate error
is 878.59
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Example Production Optimization
Repeating more iterations, the following values
are obtained
Iteration x1 x2 x3
1 2 3 4 5 6 952.38 1525.5 1314.1 1474.5 1406.0 1451.9 5 37.570 16.085 10.874 4.8686 3.1618 1601.8 1379.8 1548.2 1476.3 1524.5 1501.9 37.570 16.085 10.874 4.8686 3.1618 1.5021 257.32 26.295 89.876 27.694 49.863 32.554 61.138 878.59 70.743 224.53 44.459 53.170
! Notice After six iterations, the absolute
relative approximate errors are decreasing, but
are still high.
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Example Production Optimization
Repeating more iterations, the following values
are obtained
Iteration x1 x2 x3
20 21 1439.8 1439.8 0.00064276 0.00034987 1511.8 1511.8 0.00034987 0.00019257 36.115 36.114 0.0091495 0.0049578
The value of closely approaches the true
value of
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Gauss-Seidel Method Pitfall
Even though done correctly, the answer may not
converge to the correct answer This is a pitfall
of the Gauss-Siedel method not all systems of
equations will converge.
Is there a fix?
One class of system of equations always
converges One with a diagonally dominant
coefficient matrix.
Diagonally dominant A in A X C is
diagonally dominant if
for all i and
for at least one i
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Gauss-Seidel Method Pitfall
Diagonally dominant The coefficient on the
diagonal must be at least equal to the sum of the
other coefficients in that row and at least one
row with a diagonal coefficient greater than the
sum of the other coefficients in that row.
Which coefficient matrix is diagonally dominant?
Most physical systems do result in simultaneous
linear equations that have diagonally dominant
coefficient matrices.
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Gauss-Seidel Method Example 2
Given the system of equations
The coefficient matrix is


With an initial guess of
Will the solution converge using the Gauss-Siedel
method?
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Gauss-Seidel Method Example 2
Checking if the coefficient matrix is diagonally
dominant

The inequalities are all true and at least one
row is strictly greater than Therefore The
solution should converge using the Gauss-Siedel
Method
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Gauss-Seidel Method Example 2
Rewriting each equation
With an initial guess of





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Gauss-Seidel Method Example 2
The absolute relative approximate error




The maximum absolute relative error after the
first iteration is 100
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Gauss-Seidel Method Example 2
After Iteration 1

Substituting the x values into the equations
After Iteration 2

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Gauss-Seidel Method Example 2
Iteration 2 absolute relative approximate error


The maximum absolute relative error after the
first iteration is 240.61 This is much larger
than the maximum absolute relative error obtained
in iteration 1. Is this a problem?
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Gauss-Seidel Method Example 2
Repeating more iterations, the following values
are obtained
Iteration a1 a2 a3
1 2 3 4 5 6 0.50000 0.14679 0.74275 0.94675 0.99177 0.99919 100.00 240.61 80.236 21.546 4.5391 0.74307 4.9000 3.7153 3.1644 3.0281 3.0034 3.0001 100.00 31.889 17.408 4.4996 0.82499 0.10856 3.0923 3.8118 3.9708 3.9971 4.0001 4.0001 67.662 18.876 4.0042 0.65772 0.074383 0.00101
The solution obtained is
close to the exact solution of .
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Gauss-Seidel Method Example 3
Given the system of equations
Rewriting the equations


With an initial guess of
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Gauss-Seidel Method Example 3
Conducting six iterations, the following values
are obtained
Iteration a1 A2 a3
1 2 3 4 5 6 21.000 -196.15 -1995.0 -20149 2.0364105 -2.0579105 95.238 110.71 109.83 109.90 109.89 109.89 0.80000 14.421 -116.02 1204.6 -12140 1.2272105 100.00 94.453 112.43 109.63 109.92 109.89 50.680 -462.30 4718.1 -47636 4.8144105 -4.8653106 98.027 110.96 109.80 109.90 109.89 109.89
The values are not converging. Does this mean
that the Gauss-Seidel method cannot be used?
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Gauss-Seidel Method
The Gauss-Seidel Method can still be used
The coefficient matrix is not diagonally dominant
But this is the same set of equations used in
example 2, which did converge.
If a system of linear equations is not diagonally
dominant, check to see if rearranging the
equations can form a diagonally dominant matrix.
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Gauss-Seidel Method
Not every system of equations can be rearranged
to have a diagonally dominant coefficient matrix.
Observe the set of equations
Which equation(s) prevents this set of equation
from having a diagonally dominant coefficient
matrix?
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Gauss-Seidel Method
Summary

• Advantages of the Gauss-Seidel Method
• Algorithm for the Gauss-Seidel Method
• Pitfalls of the Gauss-Seidel Method

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Gauss-Seidel Method
Questions?
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Additional Resources

• For all resources on this topic such as digital
  audiovisual lectures, primers, textbook chapters,
  multiple-choice tests, worksheets in MATLAB,
  MATHEMATICA, MathCad and MAPLE, blogs, related
  physical problems, please visit
• http//numericalmethods.eng.usf.edu/topics/gauss_s
  eidel.html

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